On the cohomology ring of symplectic fillings

Abstract

We consider symplectic cohomology twisted by sphere bundles, which can be viewed as an analogue of symplectic cohomology with local systems. Using the associated Gysin exact sequence, we prove the uniqueness of part of the ring structure on cohomology of fillings for those asymptotically dynamically convex manifolds with vanishing property considered by Zhou (Int. Math. Res. Not. 2020 (2020) 9717-9729 and J. Topol. 14 (2021) 112-182). In particular, for any simply connected 4nC1dimensional flexibly fillable contact manifold Y , we show that the real cohomology H .W / is unique as a ring for any Liouville filling W of Y as long as c 1 .W / D 0. Uniqueness of real homotopy type of Liouville fillings is also obtained for a class of flexibly fillable contact manifolds.